Embed Size (px)
Citation preview
Pergamon M~hanlcs Rosearch Comnmnicationa, Vol. 21, No. 3, pp. 289-296, 1994
Copyright • 1994 Elaevier Scie~e Ltd P6n~d in the USA. All ~m m~rved
0093-6413/94 $6.00 + .00
LAMINAR FREE CONVECTION FLOW OVER A CONE EMBEDDED IN A STRATIFIED MEDIUM
m
R.K. Tripathi, A. Sau and G. Nath
Department of Mathematics, Indian Institute of Science
Bangalore-560 012, India.
('Received 7 September 1993; accepted for print 25 January 1994)
1 Introduction
Free convection flows which arise in a thermally stratified medium are of
practical importance. Interest lies in the effect of stratification on
temperature and velocity fields that arise in the flow and the consequent
heat transfer from a heated body. In many engineering applications cone-
shaped bodies are often used. Free convection flow over a vertical cone has
been studied by Hering and Crosh [I], |feting [2], tin and Chao [3], L%n [4],
AlamElr [5] and recently by W~tanabe [6]. A literature review reveals that
the free convection flow over a vertical cone immersed in a stratified medium
has not been studied so far. This has motivated us to carry out the present
analysis. Laminar free convection flows over bodies of different shapes like
vertical plate, sphere and cylinder in a stratified medium have been studied
by various investigators; see, for example, Chen and Eichhorn [?-8].
The purpose of the present work is to study a non-similar free
convection boundary layer flow over a vertical cone embedded in a stratified
medium. Non-slmilarity is purely due to stratification of the medium. The
coupled non-linear partial differential equations Eoverning the flow are flrst
llnearized usinE quasilinearlzation method and then solved numerically, using
an implicit finite difference scheme. Numerlcal results are ~[ven ('or the
velocity profile, temperature profile, skin friction parameter" and heat
transfer coefficient for two Prandtl numbers 0.72 and 6.0 which apply for air
and water, respectively, at normal temperature.
" Author to whome correspondence to be addressed.
289
290
2 Governing e q u a t i o n s
R.K. TRIPATHI, A. SAU and G. NATH
As s h o w n i n t h e i n s e t o f F i g . 1, a s t e a d y f r e e c o n v e c t i o n b o u n d a r y l a y e r
f l o w o v e r a c o n e ( a p e x a n g l e 2~') i n a l i n e a r l y s t r a t i f i e d e n v i r o n m e n t i~;
c o n s i d e r e d • T h e c o o r d i n a t e s y s t e m i s s u c h t h a t x m e a s u / - e s t h e d i s t a n c ~ ;~l()n~
t h e s u r f a c e o f t h e b o d y f r o m t h e a p e x , x=O b e i n g t h e l e a d i n g e d g e a n d y
m e a s u r e s d i s t a n c e n o r m a l l y o u t w a r d , u a n d v a r e t h e c o m p o n e n t ~ o f [ ' l u i d
v e l o c i t y i n t h e x a n d y d i r e c t i o n s , r e s p e c t i v e l y . T h e b o d y i!:; h e l d a t ~!
c o n s t a n t t e m p e r a t u r e T a n d t h e t e m p e r a t u r e of" t h e f l u i d f a r f r o m t h e b{)dy W
s u r f a c e i s g i v e n b y T m ( x ) = T m o + a x , w h e r e a = ( d T m / d x ) >0 a n d x ( : x c o s T ) i s
m e a s u r e d f r o m t h e a p e x o f t h e c o n e a n d i s p a r a l l e l t o t h e d i r e c t i o n o f t h e
g r a v i t y ( s e e F i g . l ) a n d T i s t i~e t e m p e r a t u r e o f t h e s u r r o u n d i n g a I x=O ( i . c , mO
x = 0 ) . We a l s o a s s u m e t h a t . T > T W 000
T h e e f f e c t o f" s t r a t i f l c a l i o n ( . ' a l l be e x ] ~ ) r ' ( . ' ~ q e ( t In I o l ' l n 5 ; of" ;* ~ : l l : * l } t i ( ' ; , I i,~I~
parameter S, defined as, S=aL cos~/kT where [. is the slant height of ti~e cone m
a n d AT m is the t e m p e r a t u r e d i f f e r e n c e , Tw-'l 'm, a t m i d h e i g h t o f the , b o d y . W i l h
t h e B o u s s i n e s q a s s u m p t i o n t h e b o u n d a r y l a y e r e q u a t i o n s g o v e r n i n g I h(~ !-;1 ( ;~(iy
free c o n v e c t i o n f l o w o v e r a v o r t i c a l c o n ( ~ c a n bt~ e x p i ' o [ ; . q e d ~xG :
a ( u r ) 8 ( v r ) ---- + - 0 {1} ax ay
Ou 8u 02u u ~ + v ~ = v - - + g /~( ' I ' -T m) c o s ~ (2) y @y2
3T 8T 0 2 T u y x + V~y = ~x - ~ (U)
8y
w h e r e r i s t h e r a d i u s o f n o r m a l s e c t i o n of ' t h e c o n e a t l o c a t i o n x H,nl I,, }~, , , ,
a r e k i n e m a t i c v i s c o s i t y , g r a v i t a t i o n a l a c c e l e r a t i o n , t h e r m a l d i f l ' u s i l i v i t y
a n d c o e f f i c i e n t o f t h e r m a l e x p a n s i o n , r e s p e c t i v e l y .
The boundary conditions a r e :
x >- O: u ( x ,O) = v ( x ,O ) = O, T (x ,O) : Tw }
u(x,~o) ---> O, ] '(x,oo) -~ 'I" (x ) (4) co
x < O: u ( x , y ) = 0 , T ( x , y ) : T00(x) f o r a l l y
where T (x) = T + ax cos3". The conditions for x<O are meant to avoid any CO CO0
FREE CONVECTION OVER A CONE 291
mathematical singularity. The subscripts w and m denote conditions at the
wall and at a large distance from the surface, respectively.
In regard to the present work it must be mentioned here that the validity
of the assumptions made in deriving the Eqns. (I) -(4) holds good if the
boundary layer Is assumed thin compared to the cone radius, thus neglecting
the effect of curvature [I]. Also, the normal component, of buoyancy and the
motion pressure are neglected.
Now we introduce the following nondimenslonal variables :
1/4 .. X = x / L , R = r / L , Y = y u r L / L , P r = via
U = u L / v C r l / 2 k ' V = vL/vGr I/4,L G = ( T - T ) / ( T w - T o) ( 5 )
Gr k = g ~ ( T w - T o) L 3 / u 2
w h e r e Gr k a n d P r a r e G r a s h o f n u m b e r a n d P r a n d t l n u m b e r , r e s p e c t i v e l y .
T h e n o n d i m e n s i o n a l c o n s e r v a t i v e e q u a t i o n s a r e t h e n o b t a i n e d a s
(RU) x + (RV)y = 0 (6)
UU X + VUy = Uyy + G cos~ (7)
UG x + 2S(2+S) -I U + VGy = Pr -I Cyy (8)
where subscripts X and Y denote derivatives with respect to the corresponding
variables.
The boundary conditions are :
X z O: U(X,O) = V(X,O) = O, G(X,O) = ] - 2S(2+S)-IX
U(X,~ ) --> O, ( ; (X ,~ ) --~ 0 (~))
X < O: U(X ,Y) = G(X,Y) = 0 f o r a l l Y
Using the approach of Lin and Chao [3] we apply following transformations
to the Eqns.(6) - (8)
6[XR 2 * ( 2 ~ ) - I / 2 * = UdX, n= RUY
~(X,Y) = C2~) ~z2 F(~,n), RU = ~y, RV = -~X
292 R.K. TRIPATHI , A. S A U and G. N A T H
o.[ + ] U = U * F ' , V = - ( 2 ~ ) - 1 / 2 R F + 2 ~ F ~ 2 ~ ( R 2 " U ' ) -~ F ' ~ X
U * * = OL/(uGP1/2) , '! )0 L oo 0
U(dU/dx] = g/~('I' w
. . . . [.i.0. ],. = U dU / d X , U = (1o)
We f i n d t h a t E q n . ( 6 ) l.~ i d e n t i c a l l y .¢;~tt I s f l e ( t a n d I<qns . ( 7 ) - ( 8 ) v ( . d u c , ' f f ,
F " ' + F F " + A ( G - F ' 2 ) = 2 ¢ (V 'F~ - F ' (F ' " )
P r - l G " + F G ' - 2 S ( 2 + S ) - 1 Q F" = 2 { ( F ' G { - t . ( G ' )
w h e r e A = 2 ( ~ / U ' ) ( d U * / d ~ ) Q = 2 ~ / ( R a L I " ) a n d F a n d G a r e ,
11 )
1 2 )
r e s p e c t i v e I y ,
n o n - d i m e n s i o n a l s t r e a m f u n c t i o n a n d n o n - d i m e n s i o n a l t e m p e r a t u r e . T h e
s u b s c r i p t ~ a n d s u p e r s c r i p t ' ( p r i m e ) d e n o t e d e r i v a t i v e s w i t h r e s p e c t t o ~ a n d
~q, r e s p e c t i v e l y , a n d [] i s a h y p o t h e t i c a l o u t e r " ~ t / ' e a m v e l o c i t y [ 'un( : t i o n ,
F o r a v e r t i c a l c i r c u l a r c o n e we h a v e t h e r e l a t i o n s R=X s i n ? a n d O*=cos?,
w h i c h o n s u b s t i t u t i n g i n E q n . ( 1 0 ) g i v e t h e e x p r e s s i o n s :
= (2/7)(2X 7 cos?')/2 sin~?, [] = (2X cos~)/2, A : 2/Z, %) = 4X, /,
I
~] = Y (7/2)/2 (cos3/2X1}/~ £(0/0~] : (2X/Y)(O/OX)
With the help of above expressions we can write Eqns. (11) and (12) o!5 :
F " ' + F F " + ( 2 / 7 ) ( G - F ' 2 ) = ( 4 X / 7 ) ( F ' F ~ - F x F " )
P p - 1 G " + F G ' - 2 S ( 2 ÷ S ) - I Q F ' = ( 4 X / 7 ) ( F ' G X - F x G ' ]
T h e b o u n d a r y c o n d i t i o n s t a k e t h e f o r m :
X ~- O: F = F ' = O, G = 1 - 2X S ( 2 + S ) -1
V ' - - ~ O, G - ~ 0
X < O: t : " ~ G = 0 F o r a l l 0
(1:~)
1 4 )
a [ z) = 0
as 7) ..... ) o0 1~)
1 / 2 1 / 4
I f we a p p l y t h e t r a n s f o r m a t i o n s 7) = ( ( M + S ) / 6 ) (~X ~ ; i n ' j / , 1 ) Y,
1 / 2 3 / 4 1 / 4
= ( 6 / ( M + 3 ) ) V (4X 3 s i n 2 ~ / 3 ] Gr- F, RU= ~ y , RV = - 0 X L
t o t i l e Eqm~. ( G ) - ( ~ ) ,
we c a n a l t e r n a t i v e l y g e t t h e e q u a t i o n s f o r a n i s o t h e r m a l m e d i u m ( i . e . , S = O / a s :
FREE CONVECTION OVER A CONE 293
F"'+ 6(M+3)-I( 3FF" - 2F'2+ G cos~ ) = 8(M+3)-IX(F~F '- FxF") (16)
PF-IG " + 18(M+3)-iF G' = 8(M+3)-IX(Gx F'- FxG') (17)
Equations (16) and (17) are identical to those of Watanabe [6]. ||ere M is the
cone angle parameter [6].
The local skln fl.lCtlon (Cf) and he~it t.ra,~sl'er" ( Nu x) coefflclent.s c~ul be
expressed as
2 m2 Cf = rw/P(v/L) Cr 3/4 = R U (2~) -I/2 F"
L w
Nu x = (Xqw/kATm)= - 7 I/2 2-3/4(1 + S/2) G~ (GP x cos~) I/4
where T w and qw are local shear stress and local heat transfer rate, respecti-
vely, at the vrall and k is the thermal conductivity..
4 R e s u l t s a n d d i s c u s s i o n
The governing Eqns.(13) and (14) under conditions (15) have been solved
numerically using an implicit finite difference scheme in combination with the
quasilinearlzatlon technique [9]. To fix the stepslzes for computation, they
have been optimized for all values of S, using Richardson extrapolation
formula. In order to assess the accuracy of the present method we have
compared our results in Table I with those of Watanabe [6] for an isothermal
medium with no surface mass transfer. For an isothermal medium with no
surf~ce mass transfer, the Eqns.(16) and (17) admit self-similarity and
consequently the right hand side of Eqns.(I6) and (17) which contains X will
not appear. The maximum difference between our results and the results of'
Watanabe [6] is found to be about I% . Moreover, our results for S=X=0 (i.e.,
for self-similar solution ) are in excellent agreement with I.in and Chno [3],
however, comparison is not shown here for the sake Of brevlty.
In Figs. I and 2 velocity profiles (F') ar'e shown for dlff'er'ent va]ue~; o["
stratification parameter S at two X-locations. It is clear that stratification
reduces the buoyancy force, which results in lower velocities. The effect of
stratification is more pronounced at higher locations. For instance, for Pr=
2 9 4 R.K. TRIPATHI, A. SAU and G. NATH
0.72, at X=0.4 the reduction in peak velocity is about 23X when the ~tratil'i~-
.~0., ation parameter increases from 0.5 to 2.0, whereas at X=0.8, it i~ about ~" "
f o r t h e s a m e c h a n g e o f s t r a t i f i c a t i o n . F r o m t h e d e f i n i t i o n (;f' .~-, '~,',~ ,h.;~'v'..'~.
t h a t f o p S = 2 t i l e t e m p e r a t u r e I l l t h e a m b i e n t e q u a l s t i l e sk i r t ' s / c ( • [¢, lnp, . I ' ; l l ill 'i! :l~
X = I , a n d f o p a v a l u e o f S > 2 , a p o r t i o n a t t h e t o p o f t i l e c o n e w i l l h a v e :~
t e m p e r a t u r e l e s s t h a n t h e a m b i e n t w h i c h i n t u r n w i l l c a u s e a f l o w F ( w ( , r s a l .
F i g s . 1 and 2 a l s o s h o w l h e ( H ' t ' e c I o]" s t r a t l t + l c a L l , m <>l~ t, . l l l l , , .r ' ;~l lit-,,
p r o f i l e a a t t w o d i f f e r e n t X l o c a t i o n s , f o r P r a n d t 1 n u m b e r s O. "72 ~md (q. !:).
S l i g h t n e g a t i v e n o n - d i m e n s i o n a l t e m p e r a t u r e s a r e f o u n d f a r " f r o m t h e s u r f a c e
f o r h i g h e r v a l u e s o f s t r a t i f i c a t i o n p a r a m e t e r . T h e o b s e r v e d t r e n d s a r e c l e a r l y
a result of the increase in the ambient temperature T with x. Since the f'luid o0
with Pr=6.0 has a lower thermal diffusivity than that of the ~']uid with
PF=O. 7 2 , t h e d i p i n l h e / e m p e r ; ~ l u r e p v o ( ' l le~; I~ m o w t ' o r I h , ' c:~s<: ,~l' t ' r - ~3. ~1
than for the case of Pr=0,72.
E f f e c t o f s t r a t i f i c a t i o n o n h e a t t r a n ~ f ' e r c o e f f ' i c l e n l al~d !~kir l t ' v i~ : l i ~
p a r a m e t e r i s s h o w n i n F i g . 3 . H e a t t r a n s f e r c o e f f i c i e n t Nu ( l o c a l N u s s t ~ l t x
n u m b e r ) i s s h o w n i l l t e r m s o f Nu ( G r c o s ~ ) -1 /4 . Fig. 3 a l s o s h o w s l h a t t h e x x
s k i n f r i c t i o n p a r a m e t e r F " r e d u c e s w i t h t h e increase o f s t r a t i f i c a t i o n . I b i s ; w
l s d u e t o t h e r e d u c t i o n i n b u o y a n c y f o r c e wi t h t h e i n c r c a . ~ e <~i' lh( : lun;~l
s t r a t i f i c a t i o n .
S Conclusions
I t I s f o u n d t h a i . v e l o c i t y i s s l g n i t ' i < : a n t l y r e d u c e d b y Ih ( . i H , ' ~ , ' : , ; , : ~ t
s t r a t i f i c a t i o n p a r a m e t e r . E f f e c t o f s t r a t i f i c a t i o n i s m o r e p r o n o u r ~ c e d a t
h i g h e r X - l o c a t i o n s . . ~ l l g h t r e v e r s a l I n t e m p e r a t u r e p r o t ' l I c ~ i~; t '~mHd I~,~-
h i g h e r s t r a t i f i c a t i o n . H e a t t r a n s f e r c o e f f i c i e n t , a n d s k i n f r i c t i o n p a r a m e t e v
a r e a f f e c t e d s i g n i f i c a n t l y b y t h e s t r a t i f i c a t i o n .
References
i. R.G. Hering and R.J. Erosh, Int. J. Heat Mass Transfer S, 1059 (1.q62).
2 . R . C . H e r i n g , I n t . J . H e a t M a s s T r a n s f e r 8 , I 3 3 3 ( 1 9 6 5 ) .
FREE CONVECTION OVER A CONE 295
3 F.N. L ln and B.T. Chao, Trans . ASME J. Heat T r a n s f e r 94, 43S (1974) .
4 F.N. Lin, Lett. Heat Mass Transfer 3, 49 (1976).
5 M. Alamglr, Trans. ASME J. Heat Transfer 101, 174 (1979)]
6 T. Watanabe, Acta Mech. 87, I (1991).
7 C.C. Chen and R. Eichhorn, Trans. ASME J. Heat Transfer 98, 446 (1978).
8 C.C. Chen and R. Eichhorn, Trans. ASME J. }{eat Transfer 101, 566 (1979).
9 K. Inouye and A. Tare, AIAA J. 12, 5~8 (19'74).
TABLE 1
Comparison of the heat transfer parameter and skin
friction parameter for Pr = 0.?3, S = O.
Watanabe [ 6 ] P r e s e n t r e s u l t s
F" -C' F" -C' W W W W
0.11556 (~=30 °) 0 .84043 0.67894 0.84613 0.67512
0 .24503 (~=45 °) 0 .70734 0.63238 0.71105 0.62441
0 .42410 (~=60 ° ) 0 .63098 0.56463 0.53810 0.56734
II
b-
0.3
o ,..H 0.12
H
Pr = 0.72 .~
0 2.0 4.0 6.0 8.0
0
x" 5~ ~D
FIG. 1
V e l o c i t y and t e m p e r a t u r e p r o f i l e s f o r v a r i o u s v a l u e s o f S. I n s e t :
The coordinate system; ~ , F'(~,~); --, C(~,n)
296 R.K. TRIPATHI, A. SAU and G. NATH
0.I!
"u_ ~L"
e,,a ---_ 0 .1 (
o ~ ( . 9
ID
,-5, ---- 0 . 0 5 H 2~
0 0
Pr= 6.0 I ,
- - m
= 0 4 , S = 0 . 5 O
:\/,
2 . 0 4 0 6 . 0 7/
0 5 x t _'>
FIG. 2
Velocity a nd t e m p e r a t u r e p r o f ' i l e s £or" v a r i o u s v a l u e s
o f 5; , F ' ( ~ , n ) ; , G ( ~ , ~ )
0 . , 3 6 2 . 0 ~ , ~
\ ~ "---C: 0 ~2 s \ 6 ' "- ~ ' . " -'0 5
.~4 / ~ -~-.0. " < - ' , ",, . . . . 0.12
OL . . . . I I • .... I .I l I l I : '~J 0 0 0 . 5 1,0
X
FIG. 3
L o c a l h e a t t r a n s f e r a nd l o c a l s k i n t ' r i c t i o n F e s u l t s t ' o r v a r ' i o u - ~
v a l u e s o f S; - - NUx(Gr x c o s ~ ) - l / 4 ; F'" ' W
